If $\displaystyle \mu (E_n) < \infty $ for $\displaystyle n \in \mathbb {N} $, and $\displaystyle 1_{E_n} \rightarrow f $ in $\displaystyle L^1 $, then f equals to the characteristic function of a measurable set.

Proof so far.

So I know that $\displaystyle \int \mid 1_{E_n} - f \mid d \mu \rightarrow 0 $

And I need to find a measurable set, say M, such that $\displaystyle f=1_M$

I apologize for the lack of significant amount of work here, but I just don't really know how to start here, any hints?

Thank you.